# What is the difference between conditional and posterior probability?

I'm having having understanding the difference between conditional and posterior probability.

Conditional probability:

...a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred.

Posterior probability:

...the conditional probability that is assigned after the relevant evidence or background is taken into account.

Are they essentially the same?

• Posterior probability is in a sense a Bayesian application of conditional probability disguised as likelihood, but they are not the same. Frequentist statisticians would happily use conditional probability without claiming to apply posterior probability Jun 12, 2016 at 20:55
• Looking at the descriptions of the two ideas on their wikipedia pages, it seems that Posterior probability is a special case of conditional probability, where the condition is explicitly the evidence from an experiment being used as the condition. Jun 12, 2016 at 21:04

Posterior probability is a conditional probability, but more specifically implies the probability of a particular parameter value(s) from a given parameter space when a given set of observations (say $X_i$) have been observed. So, you could say it's the revised prior for the parameter given observed $X$.

Conditional Probability: If E and F are 2 events associated with the same sample space of a random experiment, The conditional probability of event E given that F has occurred,i.e. P(E|F) = P(E,F)\P(F)

Similarly, The conditional probability of event F given that E has occurred,i.e. p(F|E) = P(E,F)\P(F)

Here, P(E,F) is the joint probability of E and F i.e Probability that both even E and F has occurred.

Let say we have the P(E|F), P(F) (and P(E) determined by the theorem of total probability) We can determine the probability of P(F|E) using Bayes rule and in this case, P(F|E) is called the posterior probability of event E, given conditional probability P(E|F) and prior P(F)